reserve n,m,k for Nat;
reserve x,X,X1 for set;
reserve r,p for Real;
reserve s,g,x0,x1,x2 for Real;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of REAL,the carrier of S;
reserve s1,s2 for Real_Sequence;
reserve Y for Subset of REAL;

theorem Th2:
for h1,h2 be PartFunc of REAL,the carrier of S
for seq be Real_Sequence
  st rng seq c= dom h1 /\ dom h2
    holds (h1+h2)/*seq = h1/*seq + h2/*seq & (h1-h2)/*seq = h1/*seq - h2/*seq
proof
   let h1,h2 be PartFunc of REAL,the carrier of S;
   let seq be Real_Sequence;
A1:dom h1 /\ dom h2 c= dom h1 & dom h1 /\ dom h2 c= dom h2
     by XBOOLE_1:17;
   assume A2: rng seq c= dom h1 /\ dom h2;
   now let n be Nat;
A3: n in NAT by ORDINAL1:def 12;
A4: h1/.(seq.n) = (h1/*seq).n & h2/.(seq.n) = (h2/*seq).n
      by A1,A2,FUNCT_2:109,XBOOLE_1:1,A3;
A5: rng seq c= dom (h1 + h2) by A2,VFUNCT_1:def 1; then
A6: seq.n in dom (h1 + h2) by Th1;
    thus ((h1+h2)/*seq).n = (h1+h2)/.(seq.n) by A5,FUNCT_2:109,A3
      .= (h1/*seq).n + (h2/*seq).n by A4,A6,VFUNCT_1:def 1;
   end;
   hence (h1+h2)/*seq = h1/*seq+h2/*seq by NORMSP_1:def 2;
   now let n be Nat;
A7: n in NAT by ORDINAL1:def 12;
A8: h1/.(seq.n) = (h1/*seq).n & h2/.(seq.n) = (h2/*seq).n
      by A1,A2,FUNCT_2:109,XBOOLE_1:1,A7;
A9: rng seq c= dom (h1 - h2) by A2,VFUNCT_1:def 2; then
A10: seq.n in dom (h1 - h2) by Th1;
    thus ((h1-h2)/*seq).n = (h1-h2)/.(seq.n) by A9,FUNCT_2:109,A7
      .= (h1/*seq).n - (h2/*seq).n by A8,A10,VFUNCT_1:def 2;
   end;
   hence thesis by NORMSP_1:def 3;
end;
